Complete Integral Closure and Noetherian Property for Integer-Valued Polynomial Rings

Denote by A′ the integral closure of the ring A and by A* its complete integral closure. Let D be a domain with quotient field K and E ⊆ K be a subset. We study the integer-valued polynomial rings Int(E, D)′, Int(E, D)*, Int(E, D′), Int(E, D*) and the relations of their properties with the noetherianity of Int(E, D).